Question: All of the 5th grade teachers and students from Loyola went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$61.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$25.50$ each for teachers and $$9.50$ each for students, and the group paid $$187.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 61}$ ${25.5x+9.5y = 187.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-9y = -183}$ ${25.5x+9.5y = 187.5}$ Add the top and bottom equations together. $ 0.5y = 4.5 $ $ y = \dfrac{4.5}{0.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {8.5x+3y = 61}$ to find $x$ ${8.5x + 3}{(9)}{= 61}$ $8.5x+27 = 61$ $8.5x = 34$ $x = \dfrac{34}{8.5}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {25.5x+9.5y = 187.5}$ and get the same answer for $x$ ${25.5x + 9.5}{(9)}{= 187.5}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.